I have the following problem:
Let $z_1,...,z_n$ be complex numbers such that $|z_i|=1, \forall i$. Prove that there exists $z\in \mathbb{C}$ with $|z|=1$ such that $|(z-z_1)(z-z_2)\cdot ...\cdot (z-z_n)|\geq 2$.
I could only prove that the bound is 1, using maximum modulus theorem, but I don't have any idea to prove the desired bound.
If you can help me, I'll be very grateful with you.